The Monty Hall Problem

You're on a game show. There are three doors. Behind one is a car. Behind the other two are goats. You pick a door. The host, who knows what's behind the doors, opens a different door revealing a goat. Then asks: do you want to switch to the other unopened door?

Most people say it doesn't matter. It feels like a 50/50 coin flip between two remaining doors. The math says otherwise. Switching wins 2/3 of the time. Staying wins only 1/3.

Wait, why?

When you first pick, you have a 1/3 chance of being right. That means there's a 2/3 chance the car is behind one of the other two doors. When the host opens a goat door, that 2/3 probability doesn't split across the two remaining doors. It concentrates entirely on the one you didn't pick. The host's action gives you information, but only if you use it by switching.

Another way to think about it. If you always stay, you only win when your initial pick was right (1/3). If you always switch, you win whenever your initial pick was wrong (2/3). Your initial pick is wrong most of the time, so switching wins most of the time.

Play it yourself

This is one of those things that's hard to believe until you experience it. Play the game below and watch the stats accumulate. Try staying for a while, then switching, and see what you get.

Your Results So Far

Here's what your play has produced. If the sample is small, keep going. Or run the simulation below to see thousands of games play out instantly.

Still Not Convinced?

If it still feels wrong, you're in excellent company. When Marilyn vos Savant published this answer in 1990, nearly 1,000 people with PhDs wrote in to tell her she was wrong. The problem is genuinely hard to accept, even for people who understand probability professionally.

The key insight is that the host's action is not random. The host always opens a goat door. This constraint is what makes switching advantageous. If the host opened a door at random (and might reveal the car), then switching wouldn't help. But that's not the game.

Run the simulation a few thousand times and watch the switch win rate converge on 66.7% and the stay win rate on 33.3%. There's something satisfying about watching the numbers settle into place, even when your gut keeps insisting it should be 50/50.

The Broader Insight

The Monty Hall problem isn't really about game shows. It's about something we do all the time. We ignore new information because our first instinct feels right.

When the host opens a door, the situation has changed. New information is on the table. But most people treat the decision as if nothing happened. Two doors, coin flip, done. They anchor to their original choice and don't update.

I see this pattern in plenty of decision contexts. In hiring, we stick with our first impression of a candidate even after new evidence arrives. In investing, we hold a position because we already made the decision, even when the circumstances have shifted. In diagnosis, we anchor to the first explanation and dismiss contradicting test results.

The Monty Hall lesson: when the situation changes and you receive new information, your original choice deserves to be re-evaluated. Loyalty to a past decision, when the facts have moved, is closer to inertia than conviction.